GEN-MATH L1-L14 PDF

MIDTERMS: THIRD QUARTER REVIEWER GENERAL MATHEMATICS Lesson 1: Functions relation h is not a function because there is at least one element in X for which there A relation is a rule that relate...

MIDTERMS: THIRD QUARTER REVIEWER GENERAL MATHEMATICS Lesson 1: Functions relation h is not a function because there is at least one element in X for which there A relation is a rule that relates values is more than one corresponding y-value. from a set of values (called the domain) to For example, x=7 corresponds to y = 11 or a second set of values (called the range). 13. Similarly, x=2 corresponds to both y=17 or 19. A relation is a set of ordered pairs (x,y) A relation between two sets of numbers A function is a relation where each can be illustrated by a graph in the element in the domain is related to only Cartesian plane, and that a function one value in the range by some rule. passes the vertical line test. A function is a set of ordered pairs (x,y) What is a vertical line test? Example 1. Which of the following A graph represents a function if and only if relations are functions? each vertical line intersects the graph at f= {(1,2),(2,3),(3,5),(4,7)} most once. g= {(1,3),(1,4),(2,5),(2,6),(3,7)} Example 3. Which of the following can be graphs of functions? h= {(1,3),(2,6),(3,9),…,(m,3n),…} Solution.The relations f and h are functions because no two ordered pairs have the same x-value but different y- values. Meanwhile, g is not a function because (1,3) and (1,4) are ordered pairs with the same x-value but different y- values. Relations and functions can be represented by mapping diagrams Example 2. Which of the following mapping diagrams represent functions? Solution. Graphs a.), b.), c.) are graphs of functions while d.) and e.) are not because they do not pass the vertical line test. Functions as representations of real- Solution. The relations f and g are life situations. functions because each value y in Y is unique for a specific value of x. The Functions can often be used to model real situations. Identifying an appropriate PALAWAN NATIONAL SCHOOL SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS STUDENTS’ ASSOCIATION (STEMSA) Bgy. Manggahan, Puerto Princesa City, Palawan Telephone No.: PNS STEMSA 09684548804 E-mail Address: [email protected]  MIDTERMS: THIRD QUARTER REVIEWER GENERAL MATHEMATICS functional model will lead to a better understanding of various phenomena. Example 5. Give a function C that can represent the cost of buying x meals, if Example 8. A jeepney ride costs P8.00 for one meal costs P40. the first 4 kilometers, and each additional integer kilometer adds P1.50 to the fare. Solution. Since each meal costs P40, Use a piecewise function to represent the then the cost function is C(x) = 40x. jeepney fare in terms of the distance (d) in Example 6. One hundred meters of kilometers. fencing is available to enclose a Solution. The input value is distance and rectangular area next to a river (see the output is the cost of the jeepney fare. If figure). Give a function A that can F(d) represents the fare as a function of represent the area that can be enclosed, distance, the function can be represented in terms of x. as follows. Lesson 2: Evaluating Functions Solution. The area of the rectangular Evaluating a function means replacing enclosure is A = xy. We will write this as a the variable in the function, in this case x, function of x. Since only 100 m of fencing with a value from the function's domain is available, then x + 2y = 100 or y = (100 and computing for the result. To denote – x)/2 = 50 – 0.5x. Thus, A(x) = x(50 – that we are evaluating f at a for some a in 0.5x) = 50x – 0.5x2. the domain of f, we write f(a). Piecewise functions. Example 1. Evaluate the following Some situations can only be described by functions at x = 1.5: more than one formula, depending on the value of the independent variable. Example 7. A user is charged P300 monthly for a particular mobile plan, which includes 100 free text messages. Messages in excess of 100 are charged Solution. Substituting 1.5 for x in the P1 each. Represent the monthly cost for functions above, we have text messaging using the function t(m), where m is the number of messages sent in a month. Solution. The cost of text messaging can be expressed by the piecewise function: PALAWAN NATIONAL SCHOOL SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS STUDENTS’ ASSOCIATION (STEMSA) Bgy. Manggahan, Puerto Princesa City, Palawan Telephone No.: PNS STEMSA 09684548804 E-mail Address: [email protected]  MIDTERMS: THIRD QUARTER REVIEWER GENERAL MATHEMATICS Example 2. Find g( 4) and r(1) where g Multiplication: and r are as defined in the previous (a) Rewrite the numerator and example. denominator in terms of its prime factors. Solution. This is not possible because 4 (b) Common factors in the numerator and is not in the domain of g(x) and 1 is not in denominator can be simplified as “1” (this the domain of r(x). is often called “cancelling”). Example 3. Evaluate the following (c) Multiply the numerators together to get functions, where f and q are as defined in the new numerator. Example 1. (d) Multiply the denominators together to get the new denominator. Example 3. Find the product of 10/21 and 15/8. Use cancellation of factors when convenient. Lesson 3: Operations on Solution. Express the numerators and Functions denominators of the two fractions into their prime factors. Multiply and cancel out Addition and Subtraction: common factors in the numerator and the (a) Find the least common denominator denominator to reduce the final answer to (LCD) of both fractions. lowest terms. (b) Rewrite the fractions as equivalent fractions with the same LCD. (c) The LCD is the denominator of the Example 4. Find the product of resulting fraction. (d) The sum or difference of the numerators is the numerator of the Solution. Express the numerators and resulting fraction. denominators of the two rational Example 1. Find the sum of 1/3 and 2/5. expressions into their prime factors. Multiply and cancel out common factors in Solution. The LCD of the two fractions is the numerator and the denominator to 15. reduce the final answer to lowest terms. Note the similarity in the process between this example and the previous one on fractions. Example 2. Find the sum of 1/x-3 and 2/x- 5 PALAWAN NATIONAL SCHOOL SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS STUDENTS’ ASSOCIATION (STEMSA) Bgy. Manggahan, Puerto Princesa City, Palawan Telephone No.: PNS STEMSA 09684548804 E-mail Address: [email protected]  MIDTERMS: THIRD QUARTER REVIEWER GENERAL MATHEMATICS Division: Example 5. Determine the following functions. To divide two fractions or rational expressions, multiply the dividend with the reciprocal of the divisor. Example 5.Divide Solution. Definition: Example 6. Express the following Let f and g be functions. functions as the sum, difference, quotient, or product of 1. Their sum, denoted by , is the function denoted by (f+g)(x)=f(x)+g(x). 2. Their difference, denoted by , is the function denoted by (f-g)(x)=f(x)-g(x).. 3. Their product, denoted by , is the function denoted by (f g)(x)=f(x) g(x).. 4. Their quotient, denoted by , is the function denoted by (f/g)(x)=f(x)/g(x). , Solution. excluding the values of x where g(x)=0. Use the following functions below for Example 5. Lesson 4: Representing Real- Life Situations Using Rational Functions A polynomial function p of degree n is a function that can be written in the form PALAWAN NATIONAL SCHOOL SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS STUDENTS’ ASSOCIATION (STEMSA) Bgy. Manggahan, Puerto Princesa City, Palawan Telephone No.: PNS STEMSA 09684548804 E-mail Address: [email protected]  MIDTERMS: THIRD QUARTER REVIEWER GENERAL MATHEMATICS can be used to determine the exact value at which the maximum occurs). After 1 hour, the graph suggests that drug concentration decreases until it is almost zero. A rational function is a function form f(x0=p(x)/q(x) where p(x) and q(x) are Lesson 5: Rational Functions, polynomial functions and q(x) is not the Equations, and Inequalities zero function (i.e., q(x)≠0).The domain of A rational expression is an expression f(x) is the set of all values of x where that can be written as a ratio of two q(x)≠0. polynomials. Example 1. An object is to travel a The definitions of rational equations, distance of 10 meters. Express velocity v inequalities, and functions are shown as a function of travel time t, in seconds. below. Solution. The following table of values show v for various values of t. The function v(t)=10/t can represent v as a function of t. Example 2. Suppose that c(t)=5t/t^2+1(in mg/mL) represents the concentration of a drug in a patient's bloodstream t hours A rational equation or inequality can be after the drug was administered. Construct solved for all x values that satisfy the a table of values for c(t) for t = 1, 2, 5, 10. equation or inequality. A rational function Round off answers to three decimal places. expresses a relationship between two Use the table to sketch a graph and variables (such as x and y), and can be interpret the results represented by a table of values or a Solution. graph. Lesson 6: Solving Rational Equations and Inequalities To solve a rational equation: (a) Eliminate denominators by multiplying each term of the equation by the least common denominator. (b) Note that eliminating denominators may introduce extraneous solutions. Check the solutions of the transformed equations with the original equation. The graph indicates that the maximum drug concentration occurs around 1 hour Example 1. Solve for x:2/x-3/2x=1/5 after the drug was administered (calculus PALAWAN NATIONAL SCHOOL SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS STUDENTS’ ASSOCIATION (STEMSA) Bgy. Manggahan, Puerto Princesa City, Palawan Telephone No.: PNS STEMSA 09684548804 E-mail Address: [email protected]  MIDTERMS: THIRD QUARTER REVIEWER GENERAL MATHEMATICS Solution. The LCD of all the the numerator and denominator is a useful denominators is 10x. Multiply both sides of strategy). the equation by 10x and solve the ii. Mark the numbers found in (i) on a resolving equation. number line. Use a shaded circle to Example 2. Solve for x: indicate that the value is included in the solution set, and a hollow circle to indicate that the value is excluded. These numbers partition the number line into intervals. iii. Select a test point within the interior of each interval in (ii). The sign of the rational Solution. Factor each denominator in the expression at this test point is also the rational expression. sign of the rational expression at each interior point in the aforementioned interval. iv. Summarize the intervals containing the solutions. Multiply the LCD to both sides of the equation to remove the denominators. Example 3. Solve the inequality 2x/x+1≧1. Solution. Upon reaching this step, we can use (a) Rewrite the inequality as a single strategies for solving polynomial function. rational expression. Since x=-2makes the original equat ion (b) The rational expression will be zero for undefined,x=5 is the only solution. x = 1 and undefined for x = –1. The value x = 1 is included while x = –1 is not. Mark To solve rational inequalities: these on the number line. Use a shaded circle for x = 1 (a solution) and an (a) Rewrite the inequality as a single unshaded circle for x = -1 (not a solution). rational expression on one side of the inequality symbol and 0 on the other side. (b) Determine over what intervals the (c) Choose convenient test points in the rational expression takes on positive and intervals determined by –1 and 1 to negative values. determine the sign of x-1/x+1 in these i. Locate the x values for which the rational intervals. Construct a table of signs as expression is zero or undefined (factoring shown below. PALAWAN NATIONAL SCHOOL SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS STUDENTS’ ASSOCIATION (STEMSA) Bgy. Manggahan, Puerto Princesa City, Palawan Telephone No.: PNS STEMSA 09684548804 E-mail Address: [email protected]  MIDTERMS: THIRD QUARTER REVIEWER GENERAL MATHEMATICS (d) Summarize the intervals satisfying the inequality. The solution set of the inequality is the set {x∈R|x 0, b ≠ 1). The definitions of exponential equations, inequalities and functions are shown below. Determine whether the given is an exponential function, an exponential equation, an exponential inequality, or none of these. 1. f(x) = 5x 2 (Answer: None of these) 2. 2 ≥ (1/2) x (Answer: Exponential inequality) 3. 74x = y (Answer: Exponential function) 4. 4(10^x - 2 ) = 500 (Answer: Exponential equation) 5. 7 < 14 ^x + 3 (Answer: Exponential inequality) 6.y = 0.5x Property of Exponential Inequalities (Answer: Exponential function) Lesson 14: Solving Exponential Equations and Inequalities One-to-one Property of Exponential Functions PALAWAN NATIONAL SCHOOL SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS STUDENTS’ ASSOCIATION (STEMSA) Bgy. Manggahan, Puerto Princesa City, Palawan Telephone No.: PNS STEMSA 09684548804 E-mail Address: [email protected] MIDTERMS: THIRD QUARTER REVIEWER GENERAL MATHEMATICS PALAWAN NATIONAL SCHOOL SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS STUDENTS’ ASSOCIATION (STEMSA) Bgy. Manggahan, Puerto Princesa City, Palawan Telephone No.: PNS STEMSA 09684548804 E-mail Address: [email protected] MIDTERMS: THIRD QUARTER REVIEWER GENERAL MATHEMATICS PALAWAN NATIONAL SCHOOL SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS STUDENTS’ ASSOCIATION (STEMSA) Bgy. Manggahan, Puerto Princesa City, Palawan Telephone No.: PNS STEMSA 09684548804 E-mail Address: [email protected]